Error estimates in the Fast Multipole Method for scattering problems Part 2: Truncation of the Gegenbauer series

Abstract

Abstract.We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, e i| u− v| 4πi| u− v| , which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices ≤ L. We prove that if v = | v| is large enough, the truncated series gives rise to an error lower than as soon as L satisfieswhere W is the Lambert function, K(α) depends only on α = | u| | v| and C , δ, γ are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates of the error in the fast multipole method for scattering computation.Mathematics Subject Classification. 33C10, 33C55, 41A80.

Installation: numerical investigation

“…L is a cut-off value which controls the accuracy of the approximation and must be properly determined, cf. [15,14]. The argument of the spherical Hankel function in eqn.…”

Validation of a Model for Open Rotor Noise Predictions and Calculation of Shielding Effects using a Fast BEM

“…The numerical expenditures have been shifted into the so-called transfer function M L,ŝ (c − a), which is a finite sum over products of Hankel functions and Legendre polynomials and depends for a fixed point on the sphere only on the difference c − a. The limit lim L→∞ M L,ŝ (u) is divergent and the cutoff value L must be chosen carefully [22]. Unfortunately, it is not possible here to discuss this topic further.…”